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JEE Advanced

Mole fraction of urea in 900 gram water is 0.05 density of solution is $1.2g/cm_{3}$. Find the molarity of solution.

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Let $P_{1}:2x+y−z=3$and $P_{2}:x+2y+z=2$be two planes. Then, which of the following statement(s) is (are) TRUE?The line of intersection of $P_{1}$and $P_{2}$has direction ratios $1,2,−1$(b) The line $93x−4 =91−3y =3z $is perpendicular to the line of intersection of $P_{1}$and $P_{2}$(c) The acute angle between $P_{1}$and $P_{2}$is $60o$(d) If $P_{3}$is the plane passing through the point $(4,2,−2)$and perpendicular to the line of intersection of $P_{1}$and $P_{2}$, then the distance of the point $(2,1,1)$from the plane $P_{3}$is $3 2 $

Let $f:R→(0,∞)$ and $g:R→R$ be twice differentiable functions such that f" and g" are continuous functions on R. suppose $f_{prime}(2)=g(2)=0,f(2)=0$ and $g_{′}(2)=0$, If $x→2lim f_{′}(x)g_{′}(x)f(x)g(x) =1$ then

Let $f:[a,b]1,∞ $be a continuous function and let $g:RR$be defined as$g(x)={0ifxbThen$$g(x)$is continuous but not differentiable at a$g(x)$is differentiable on $R$$g(x)$is continuous but nut differentiable at $b$$g(x)$is continuous and differentiable at either $a$or $b$but not both.

A line $l$ passing through the origin is perpendicular to the lines $l_{1}:(3+t)i^+(−1+2t)j^ +(4+2t)k^,∞<t<∞,l_{2}:(3+s)i^+(3+2s)j^ +(2+s)k^,∞<t<∞$ then the coordinates of the point on $l_{2}$ at a distance of $17 $ from the point of intersection of \displaystyle{l}&{l}_{{1}} is/are:

Column 1,2 and 3 contains conics, equations of tangents to the conics and points of contact, respectively.Column I, Column 2, Column 3I, $x_{2}+y_{2}=a$, (i), $my=m_{2}x+a$, (P), $(m_{2}a ,m2a )$II, $x_{2}+a_{2}y_{2}=a$, (ii), $y=mx+am_{2}+1 $, (Q), $(m_{2}+1 −ma ,m_{2}+1 a )$III, $y_{2}=4ax$, (iii), $y=mx+a_{2}m_{2}−1 $, (R), $(a_{2}m_{2}+1 −a_{2}m ,a_{2}m_{2}+1 1 )$IV, $x_{2}−a_{2}y_{2}=a_{2}$, (iv), $y=mx+a_{2}m_{2}+1 $, (S), $(a_{2}m_{2}+1 −a_{2}m ,a_{2}m_{2}+1 −1 )$If a tangent to a suitable conic (Column 1) is found to be $y=x+8$and its point of contact is (8,16), then which of the followingoptions is the only CORRECT combination?(III) (ii) (Q) (b) (II) (iv) (R)(I) (ii) (Q) (d) (III) (i) (P)

Let $f:R→R$be a differentiable function with $f(0)=0$. If $y=f(x)$satisfies the differential equation $dxdy =(2+5x)(5x−2)1 $, then the value of $(lim)_{x→∞}f(x)$is ______

The common tangents to the circle $x_{2}+y_{2}=2$ and the parabola $y_{2}=8x$ touch the circle at $P,Q$ andthe parabola at $R,S$. Then area of quadrilateral $PQRS$ is

The option(s) with the values of a and L that satisfy the following equation is (are) $∫_{0}e_{t}(sin_{6}at+cos_{4}at)dt∫_{0}e_{t}(sin_{6}at+cos_{4}at)dt =L$